Self Wright

Tag: mathematics

A comment on programming and math …

Original source here

There is another school which has not been very well represented in the literature over the years, but which has undoubtedly produced a greater positive impact on the economy. Call it the “Fortran school of programming,” which I think is well summarized by Dr. Adam Rosenberg, the self-described last buffalo of industrial mathematics. Rather than viewing mathematics as an advanced tool reserved for extremely specialized computer applications, Fortran-school programmers view the computer as an advanced tool for doing mathematics. Historically, Fortran-school programmers have tended to work in industry or in the more technically inclined parts of the government (NASA, Defense, etc.). They are often mocked by Lisp programmers for their ignorance of recursion. (For a satirical essay, see Real Programmers Don’t Use PASCAL; if you think this satire is unfair, see Dr. Rosenberg’s Style Guide.)

The mockery, I think, ought to run with at least as much force in the opposite direction. In contrast to the Fortran tradition (which proudly “sent a man to the moon” and implemented critical infrastructure in banking, communications, and so on), the culture of Lisp is almost willfully ignorant of mathematics. This ignorance is disguised by all the talk of formalism and the instinctive genuflection before the Lambda Calculus, which, not unlike the Summa Theologica, is a closed computational universe that sheds little light on the observed world.

The trouble with the Lisp-hacker tradition is that it is overly focused on the problem of programming — compilers, abstraction, editors, and so forth — rather than the problems outside the programmer’s cubicle. I conjecture that the Lisp-school essayists — Raymond, Graham, and Yegge — have not “needed mathematics” because they spend their time worrying about how to make code more abstract. This kind of thinking may lead to compact, powerful code bases, but in the language of economics, there is an opportunity cost. If you’re thinking about your code, you’re not thinking about the world outside, and the equations that might best describe it.

Although the early years in the twenty-first century seem to be favoring the Lisp-school philosophy, I predict the balance of the century will belong to the Fortran-school programmers who are able to successfully apply mathematics to practical problems. It is tempting to declare that most programming problems “don’t need math”, but this is only true in the same sense that manufacturing, or supply-chain management, or baseball, “doesn’t need math”: advanced mathematics seems completely unnecessary to existing practitioners, but only until someone figures out that a particular mathematical concept is the right way to think about the problem at hand. After that, it is vital.

There are two reasons I am optimistic about the future of mathematics in computer programming. The first has to do with the growth in the amount of data generated by web companies (“Big Data”). With more types of data at hand, there are more equations that might be applied with utility. There is a lot of interest in advanced machine-learning techniques for this reason, but even simple statistical techniques might have prove to have at least as many applications. Mathematics applied to business data will be yield better business insights, more efficient operations, better products (e.g. recommendations), and new products (e.g. prediction services).

The second reason I am optimistic about the place of mathematics in computer programming is related: the average consumer has more data than ever before, and mathematics can help to make sense of it, or at least make it more beautiful. Application areas that were traditionally considered to be “scientific computing” (for example, Geographic Information Systems or image-processing) are now of interest to regular people who own (say) a collection of geotagged digital photographs. Instagram, for example, was built on a few equations that operated on an image’s color channels. An understanding of mathematics can help the programmer solve practical problems for users as well as provide a more pleasing experience. (To this end, you might enjoy my previous essay, Winkel Tripel Warping Trouble.)

Mathematics, in the end, does not help you understand computer programming. It is not about finding metaphors, or understanding “fundamentals” that will never be applied. Rather, mathematics is a tool for understanding phenomena in the world: the motion of the planets, the patterns in data, the perception of color, or any of a myriad things in the world that might be understood better by manipulating equations. It is the hacker’s job to figure out how to encode the insight into a piece of code that will be used over and over.

Should we return to the good old days when men programmed Fortran and everything was an array? Hardly. What we need is an infusion of applied mathematics into hacker education. Hackers raised on Lisp-school essays have grown up with only one parent. (The other parent has apparently been too busy at work.) We need examples, tutorials, and war stories wherein non-trivial mathematics are applied with success in computer programs. Although braggadocio doesn’t come naturally to most computer programmers, we need hackers to toot the horn of triumph whenever a new and interesting application of mathematics is found. We need to celebrate the spirit of scientific curiosity.

Lastly, we need the next generation of aspiring hackers to incorporate mathematics into their program of self-study. We need college students to take classes in physics, engineering, linear algebra, statistics, calculus, and numerical computing, and we need them to educate their elders who grew up ignorant of these things. With the relentless proliferation of data, and the impending extinction of the Fortran-slinging old guard, there are vast opportunities for budding mathematical hackers to make a difference in the world simply by thinking about it in a more rigorous way.

Lisp, CLOS and Math

Lisp, CLOS and Math

What, you say, mathematicians just have to use Haskell ??

Stop right there, and read this.

Technically, there’s nothing new to see here at all. Lisp is high-level and low-level. Lisp is multi-paradigm. Lisp has uniform syntax. Blah blah blah. You’ve heard it all.

Still, some people seem to require an “argument from authority” (and usually still keep looking) … in which case you might be persuaded…

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Lisp, CLOS and Math

What, you say, mathematicians just have to use Haskell ??

Stop right there, and read this.

Technically, there’s nothing new to see here at all. Lisp is high-level and low-level. Lisp is multi-paradigm. Lisp has uniform syntax. Blah blah blah. You’ve heard it all.

Still, some people seem to require an “argument from authority” (and usually still keep looking) … in which case you might be persuaded by this.

An extract (describing how defclass and defgeneric can be handily abused to yield whatever you like):

For example, a mathematician who installs under CLOS the traditional mathematical categories must organize his work as follows:

  • The defclass statements will be used to define what a “set” object is, what a “group” object is, what a “chain complex” object is, and so on.

  • The defgeneric statements will be used to define functions working on these objects, but these “functions” are traditionally called in this case functors in mathematics; therefore one defgeneric statement for the sum functor, another defgeneric for the classifying space functor, and so on.

  • Finally each generic function will have various methods to adapt the generic function to specific cases; for example the product of two objects of some category is also an object of this category with the corresponding structure to be defined. Therefore one product method for the sets, another product method for the magmas, another method for the monoids, and so on. The call-next-method and change-class functions will allow these methods to possibly refer to the less specific ones.

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